Common fixed point theorems for four mappings satisfying ψ-weakly contractive conditions

Zeqing Liu,Xiaoping Zhang,Jeong Sheok Ume,Shin Min Kang

doi:10.1186/s13663-015-0271-z

Abstract

The existence and uniqueness of common fixed points for four mappings satisfying ψ- and $(\psi, \varphi)$ -weakly contractive conditions in metric spaces are proved. Four examples are given to demonstrate that the results presented in this paper generalize indeed some well-known results in the literature.

Highlights

1 Introduction and preliminaries In, Rhoades [ ] introduced the concept of φ-weakly contractive mappings and proved the following fixed point theorem, which is a generalization of the Banach fixed point theorem
Abbas and Dorić [ ], Abbas and Khan [ ], and Dutta and Choudhury [ ] proved the following fixed and common fixed point theorems for the φ- and (ψ, φ)-weakly contractive mappings
Motivated by the results in [ – ], in this paper, we introduce the concepts of ψ- and (ψ, φ)-weakly contractive conditions relative to four mappings A, B, S and T: d(Tx, Sy) ≤ ψ Mi(x, y), ∀x, y ∈ X, ( . )

Summary

Introduction

If T and S are weakly compatible, T and S have a unique common fixed point. Establish sufficient conditions which ensure the existence and uniqueness of common fixed points for the four mappings A, B, S and T satisfying ψ- and (ψ, φ)-weakly contractive conditions, respectively, in metric spaces. 2 Common fixed point theorems Our main results are as follows.

Results

Conclusion